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            <h1 id="seo-header">『计算机组成原理』数的表示及计算</h1>
            
            
              <div class="markdown-body">
                
                <h1 id="『计算机组成原理』数的表示及计算"><a href="#『计算机组成原理』数的表示及计算" class="headerlink" title="『计算机组成原理』数的表示及计算"></a>『计算机组成原理』数的表示及计算</h1><p>@[toc]</p>
<h3 id="三、数的表示及计算"><a href="#三、数的表示及计算" class="headerlink" title="三、数的表示及计算"></a>三、数的表示及计算</h3><h4 id="1-定点数的表示"><a href="#1-定点数的表示" class="headerlink" title="1.定点数的表示"></a>1.定点数的表示</h4><h5 id="1-原码表示范围："><a href="#1-原码表示范围：" class="headerlink" title="1.原码表示范围："></a>1.原码表示范围：</h5><p>符号位和数据为 i 相对独立</p>
<p><strong>N 位原码表示范围：</strong></p>
<p>二进制形式：<br>[1111….11,01111…111]</p>
<p>整数：<br>$[-(2^{n-1}-1),2^{n-1}-1]$<br>共计：$2^n-1$</p>
<p>小数：<br>$[-(1-2^{-(n-1)}),1-2^{-(n-1)}]$<br>共计：$2^n-1$</p>
<p>因为 0 有两种表示，所以比 N 为能表示的状态少 1 种。</p>
<h5 id="2-补码表示的范围"><a href="#2-补码表示的范围" class="headerlink" title="2.补码表示的范围"></a>2.补码表示的范围</h5><p><strong>N 位补码表示范围</strong></p>
<table>
<thead>
<tr>
<th></th>
<th>-4</th>
<th>-2</th>
<th>-1</th>
<th>0</th>
<th>1</th>
<th>2</th>
<th>4</th>
</tr>
</thead>
<tbody><tr>
<td>补码</td>
<td>1100</td>
<td>1110</td>
<td>1111</td>
<td>0000</td>
<td>0001</td>
<td>0010</td>
<td>0100</td>
</tr>
<tr>
<td>思考：</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td>[1000]补 的真值是谁 ?</td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
</tbody></table>
<p>是-8，也就是说补码可以表示出比原码多一位的数，而且 0 的表示方法唯一。</p>
<p><strong>考虑 N 为补码的表示形式：</strong></p>
<table>
<thead>
<tr>
<th></th>
<th>形式</th>
<th>真值 （整数）</th>
<th>真值（小数）</th>
</tr>
</thead>
<tbody><tr>
<td>最大正数</td>
<td>011111…1111</td>
<td>$2^{n-1}-1</td>
<td>$1-2^{-(n-1)}$</td>
</tr>
<tr>
<td>最大负数</td>
<td>11..11111111</td>
<td>$-2^{n-1}$</td>
<td>$-(1-2^{-(n-1)})-2^{-(n-1)}&#x3D;-1$</td>
</tr>
</tbody></table>
<h4 id="2-定点数运算"><a href="#2-定点数运算" class="headerlink" title="2.定点数运算"></a>2.定点数运算</h4><h5 id="1-加减运算："><a href="#1-加减运算：" class="headerlink" title="1.加减运算："></a>1.加减运算：</h5><p><strong>补码的加减法基本公式：</strong></p>
<p>[A+B]补&#x3D;[A]补+[B]补<br>[A-B]补&#x3D;[A]补+[-B]补<br>优点：直接运算</p>
<p><strong>溢出的判断：</strong><br>双符号位法：两个符号位同时运算，双符号位不同则溢出，上溢和下溢。</p>
<h5 id="2-移位运算"><a href="#2-移位运算" class="headerlink" title="2.移位运算"></a>2.移位运算</h5><h6 id="逻辑移位：-逻辑左移，逻辑右移"><a href="#逻辑移位：-逻辑左移，逻辑右移" class="headerlink" title="逻辑移位： 逻辑左移，逻辑右移"></a><strong>逻辑移位：</strong> 逻辑左移，逻辑右移</h6><p>移出位移走，补位位补 0</p>
<figure class="highlight"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs">eg：1010<br>左移：0100<br>右移：0101<br></code></pre></td></tr></table></figure>

<h6 id="循环移位："><a href="#循环移位：" class="headerlink" title="循环移位："></a><strong>循环移位：</strong></h6><p><strong>小循环移位</strong>小循环左移，小循环右移<br>移出移入到补位位，同时移入到标记为 C 中<br>在数字最前和最后分别有一个 C 位，储存溢出位</p>
<figure class="highlight yaml"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs yaml"><span class="hljs-string">eg：1010</span><br><span class="hljs-string">左移：1</span>  <span class="hljs-number">0101</span><br><span class="hljs-string">右移：0101</span>   <span class="hljs-number">0</span><br></code></pre></td></tr></table></figure>

<p><strong>大循环移位</strong>：大循环左移，大循环右移<br>移出位移入到 C 位，C 位移入到补位位</p>
<figure class="highlight yaml"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br></pre></td><td class="code"><pre><code class="hljs yaml"><span class="hljs-string">eg：0</span> <span class="hljs-number">1010</span><br> <span class="hljs-string">左移：1</span> <span class="hljs-number">0100</span><br> <span class="hljs-string">右移：0101</span> <span class="hljs-number">0</span><br></code></pre></td></tr></table></figure>

<h6 id="算数移位："><a href="#算数移位：" class="headerlink" title="算数移位："></a><strong>算数移位：</strong></h6><p>（1）保留数值的数学意义，左移相当于乘以 2，右移相当于除以 2<br>（2）存在溢出或精度丧失<br>原码：符号位不动，数值位逻辑左右移<br>补码：带着符号位一起移动<br>移出位移走。补位位：高位补符号位，地位补零。</p>
<h6 id="溢出过半："><a href="#溢出过半：" class="headerlink" title="溢出过半："></a>溢出过半：</h6><p>原码：符号位为 0 或 1，数值位最高位为 1<br>补码：符号位为 0，数值最高位为 1；符号位为 1，数值最高位为 0.</p>
<h4 id="3-浮点数的表示"><a href="#3-浮点数的表示" class="headerlink" title="3.浮点数的表示"></a>3.浮点数的表示</h4><p><strong>规则：</strong><br>1）组成部分、顺序<br>2）每个组成部分位数<br>3）尾数和阶码表示形式：<br>尾数：定点小数<br>阶码：定点整数</p>
<p><strong>一、IEEE754 标准</strong><br><strong>1.格式：</strong><br>32 位（float）和 64 位（double）<br><img src="https://img-blog.csdnimg.cn/20200401180114109.png" srcset="/img/loading.gif" lazyload alt="在这里插入图片描述"><br><strong>2. IEEE 有关阶码的规定</strong><br>（1）用修正过的移码表示<br>修正过的移码可以表示 0 和+-∞；</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs cpp"><span class="hljs-type">float</span>：真值（e）=机器数的值（E）<span class="hljs-number">-128</span><br><span class="hljs-type">double</span>：真值（e）=机器数的值（E）<span class="hljs-number">-1024</span><br></code></pre></td></tr></table></figure>

<p>（2）有关移码<br>$$e&#x3D;E-2^{n-1}$$</p>
<p>（3）E 位全零或者全 1 的状态都不用</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs cpp">全<span class="hljs-number">0</span>：表示数值<span class="hljs-number">0</span><br>全<span class="hljs-number">1</span>：表示+-∞<br></code></pre></td></tr></table></figure>

<p>所以正式的 IEEE754 标准：<br>$$e&#x3D;E-(2^{n-1}-1)$$</p>
<p><strong>3. IEEE 有关尾数的规定</strong><br>（1）用修正过的原码表示尾数<br>（2）最高数值位必须为 1，并且省略。<br>&#x3D;&#x3D;这样进一步提高了数据的精度。&#x3D;&#x3D;<br>（3）什么是规格化的尾数（M）<br>$$|M|&gt;&#x3D;1&#x2F;2&#x3D;(0.1)_2$$</p>
<p>因为尾数最高的数值位为 1，并且省略。<br>E：阶码<br>S：符号位<br>M：尾数<br>float：$(-1)^s<em>1.M</em>2^{E-127}$<br>double:$(-1)^s<em>1.M</em>2^{E-1023}$</p>
<p><strong>4. IEEE 与真值的相互转换</strong><br>eg:Float 类型 413C0000H 的真值是多少:</p>
<p>$413C0000H<br>&#x3D;(0100 0001 0011 1100 0000 0000 0000 0000)_2$</p>
<p>$S&#x3D;0；M&#x3D;0111；E&#x3D;1000 0010$</p>
<p>$e&#x3D;(-1)^s<em>1.M</em>2^{E-127}$<br>$&#x3D;(-1)^0<em>1.01111</em>2^{130-127}$<br>$&#x3D;(1011.11)_2$<br>$&#x3D;11.75$</p>
<p><strong>5. Float 类型的表示范围</strong><br><img src="https://img-blog.csdnimg.cn/20200401183031840.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzQ2MDIyNA==,size_16,color_FFFFFF,t_70" srcset="/img/loading.gif" lazyload alt="在这里插入图片描述"><br><strong>二、非 IEEE754 标准</strong></p>
<p><strong>1.有机组成部分</strong><br><img src="https://img-blog.csdnimg.cn/20200401183136443.png" srcset="/img/loading.gif" lazyload alt="在这里插入图片描述"><br>每部分可以用不同的编码表示，最常见的阶码和尾数都用补码表示。</p>
<p><strong>2. 表示范围</strong><br>（1）为什么要规格化？<br>&#x3D;&#x3D;为了进一步提高数据精度&#x3D;&#x3D;<br>（2）规格化的本质是什么？<br>$$|尾数|&gt;&#x3D;0.5$$</p>
<p>(3)这种本质的体现是什么？<br>尾数分别用原码和补码表示的时候规格化的尾数是什么形式的？<br>原码：数值最高位为 1<br>补码：1.0xxxxx 或者 0.1xxxxx 或者 1.1xxxxx</p>
<p>（4）IEEE754 标准如何做的规划化？<br>&#x3D;&#x3D;尾数最高的数值位为 1，并且省略。&#x3D;&#x3D;</p>
<p>设某浮点数阶码和尾数都用 4 位二进制补码表示。<br><img src="https://img-blog.csdnimg.cn/20200401183728682.png?x-oss-process=image/watermark,type_ZmFuZ3poZW5naGVpdGk,shadow_10,text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80MzQ2MDIyNA==,size_16,color_FFFFFF,t_70" srcset="/img/loading.gif" lazyload alt="在这里插入图片描述"></p>
<h4 id="4-浮点数的运算"><a href="#4-浮点数的运算" class="headerlink" title="4.浮点数的运算"></a>4.浮点数的运算</h4><p><strong>一、IEEE754 标准的加减运算</strong><br>（1）0 操作数检查<br>（2）对阶：小阶向大阶对齐，尾数右移<br>（3）尾数加减：补码双符号位<br>（4）规格化： 左规；右规</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br></pre></td><td class="code"><pre><code class="hljs cpp">左规：尾数+-不发生溢出，结果不是规格化，左规尾数不定，阶码<span class="hljs-number">-1</span>，知道规格化为止；<br>右规：尾数+-发生溢出，带两个符号位右移移位，阶码+<span class="hljs-number">1</span><br></code></pre></td></tr></table></figure>

<p>（5）舍入，阶码采用补码双符号位运算。&#x3D;&#x3D;0 舍 1 入。&#x3D;&#x3D;<br>（6）判断溢出<br>在溢出判断中，什么样的情况才算溢出？<br>&#x3D;&#x3D;阶码溢出才叫溢出&#x3D;&#x3D;<br>尾数溢出是否标志着结果的溢出？<br>&#x3D;&#x3D;尾数溢出不表示结果的溢出&#x3D;&#x3D;</p>

                
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